Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flow with Surfactants

TL;DR

This paper proves the global existence of weak solutions for a complex diffuse interface model describing two-phase flow with surfactants, coupling Navier-Stokes and Cahn-Hilliard equations with surfactant diffusion.

Contribution

It establishes the existence of weak solutions for a coupled Navier-Stokes/Cahn-Hilliard system with surfactant effects, using a novel two-step approximation method.

Findings

Existence of weak solutions is proven for the model.

The approach handles general initial data.

A two-step approximation method is developed.

Abstract

Two-phase flow of two Newtonian incompressible viscous fluids with a soluble surfactant and different densities of the fluids can be modeled within the diffuse interface approach. We consider a Navier-Stokes/Cahn-Hilliard type system coupled to non-linear diffusion equations that describe the diffusion of the surfactant in the bulk phases as well as along the diffuse interface. Moreover, the surfactant concentration influences the free energy and therefore the surface tension of the diffuse interface. For this system existence of weak solutions globally in time for general initial data is proved. To this end a two-step approximation is used that consists of a regularization of the time continuous system in the first and a time-discretization in the second step.